Let us now consider the evolution of a compact binary system in the EFT approach in the far zone. The total system formed by the radiation graviton fields \(\bar{h}_{\mu \nu }\)  and
the worldlines of the compact bodies \(\mathbf {x}_{1,2}(t)\) , is closed. However, once gravitational perturbations are integrated out, the dynamic of the worldlines is open, and one should use In-In formalism since the problem is formulated by specifying only initial conditions, as discussed in [1]}[2]}. We need to double the degrees of freedom as:
\(\bar{h}_{\mu \nu }\ \rightarrow \ (\bar{h}_{\mu \nu 1},\bar{h}_{\mu \nu 2}) \ , \qquad \mathbf {x}_k \ \rightarrow \ (\mathbf {x}_{k1},\mathbf {x}_{k2}) \ ,\) 
